Hospital Health Care: Pricing and Quality Control in a Spatial Model with Asymmetry of Information

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Hospital Health Care: Pricing and Quality Controlin a Spatial Model with Asymmetry of Information

ROSELLA LEVAGGI levaggi@eco.unibs.it

Dipartimento di Scienze Economiche, Universita di Brescia, Via S. Faustino 74/b, 25122 Brescia

The cost of hospital care depends on the quality of the service, on the personal characteristics of the patient,on the effort of the medical staff and on information asymmetry. In this article the cost minimizing propertiesof alternative payment systems will be discussed in a context where hospitals can observe patient severity andcompete according to the rules of Hotelling’s spatial competition. The scheme is designed from the standpointof a purchaser that sets up a contract with several providers for services of a given quality at the least possiblecost. Patients’ severity cannot be observed and quality cannot be verified, but the latter can be inferred through thechoice of patients. The model shows that in the health care market, prospective payments and yardstick competitionare weak instruments for cost containment; incentive compatible schemes are, at least from a theoretical pointof view, better instruments especially in a context where the purchaser can use signals relating to the variables itcannot observe. Cost inflation has two components: the information rent paid to the provider and inefficiency. Inour model the information rent is used by the provider to get more patients to his hospital; spatial competition canthen be used to curb the cost of providing hospital care.

Keywords: asymmetry of information, spatial competition, cost containment

JEL classification: I110, I180, D820

1. Introduction

Hospital care accounts for about half of total expenditure on health in most developedcountries. Cost control is therefore a major issue and several instruments have been pro-posed, ranging from pro competition measures to the use of more sophisticated contractsfor reimbursing hospital care. The general feeling is that costs, especially where prospec-tive payment systems have been adopted, have continued to increase while the level ofcompetition is still quite low and does not seem to reduce the price of the treatment.1 Thisnegative result depends on the characteristics of the market where several factors determinethis outcome irrespective of its configuration (public or private). Fiscal illusion, asymmetryof information and patients’ choices are the most important features.

In the health care market the patient has no sovereignty and it is usually assumed thatanother agent (the doctor) takes decisions on his behalf. The patient does not normally payfor each treatment received. In the private market he pays a premium that depends on theexpected cost for providing health care; in a public context health care is free at the point ofuse and it is financed through taxation. This separation creates fiscal illusion: the treatment is

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chosen on perceived quality without a proper assessment of its cost-effectiveness properties.Recent literature on the patient’s choice assigns him an active involvement, but his role isnot always welfare-improving.2

Finally, the cost of health care depends on its quality, on the effort of the medical staff andon some characteristics of the patients that are not known at the time the contract is stipulatedand that later will be observed only by the provider. As a result, the relationship betweencost, the effort of the medical staff and quality becomes unobservable. The literature haslong recognized the problem (Baron and Myerson, 1982) and offered solutions for specificcontracts (Laffont and Tirole, 1993; Laffont and Martimort, 2002). One of the most commonproblems that this literature highlights is that there might be several optimal mechanismsdesign and that their relative merits depend on the environment in which they are applied.

In this article the cost minimizing properties of alternative systems to reimburse hospitaltreatments will be discussed in an environment where the cost for health care is borne by acollective purchaser and hospitals compete for patients via quality. The scheme is designedfrom the standpoint of a purchaser (hereafter P , which might be a government body, aninsurance company, an HMO) that wants to provide hospital services at the minimum costto a population of N individuals, uniformly distributed on a line whose length is normalizedto one. Patient severity cannot be observed and quality cannot be verified, but the lattercan be indirectly controlled through the behavior of hospitals that compete for patients ala Hotelling. The model develops a two stage game. In the first stage a range of paymentschemes to attain cost minimization and the optimal quality level is defined. In the secondstage the hospitals compete for patients through quality, the purchaser anticipates hospitals’behavior and infers the amount it must pay to induce hospitals to support the optimal qualitylevel.

The paper shows that in a symmetric world the choice of the cost reimbursement ruledepends on the degree of risk aversion of the agents involved. In an asymmetrical informationframework the paper reaches the following conclusions:

• the use of prospective payments (Ellis and McGuire, 1986; Ma, 1994) is optimal only ifthe timing of information allows for true risk-sharing;

• benchmarking and yardstick competition might not be effective in reducing the extracosts caused by asymmetry of information;

• incentive compatible contracts perform better in this context, but they might be difficultto implement;

• competition on quality at the second stage permits recovery of the information rent ofthe hospital, but it does not reduce the extra cost determined by the inefficient use ofresources.

The paper will be organized as follows: in Section two we present a review of the literatureon the cost-quality trade-off; in Section three the model is sketched; in Section four thedifferent stages of the game are presented; in Section five the results of the different modelsare compared and finally Section six concludes the paper.

HOSPITAL HEALTH CARE 329

2. Cost, Quality and Asymmetry of Information

The cost of providing health care, even if the treatment is homogeneous, is uncertain becauseit depends on specific characteristics of the patient treated; for the same reason the qualityof the care provided cannot be inferred from the outcome.

This uncertainty raises the issue of risk sharing in the contract. The payment schemesrange from cost reimbursement in which the purchaser reimburses the supplier for the actualcost to prospective payment schemes, i.e. a fixed price for each person treated. If informationis symmetric, the choice of the optimal contract is determined by the degree of risk aversionof the agents involved.

In this context, Ellis and McGuire (1986) show that prospective payment systems mightbe a superior instrument and that mixed payment systems should be used when doctors arenot perfect agents. This scheme is optimal when the timing of information allows for truerisk sharing. In an asymmetry of information context, it is likely to produce cost inflation(Chalckley and Malcomson, 2002; Levaggi, 2005).

For public utilities, the literature has proposed other forms of payment such as yardstickcompetition and benchmarking where the price is set at the average observed cost for theindustry.

For hospitals, a straight application of yardstick competition might not be possible be-cause the cost of the product supplied, even within the same case mix, might have a signif-icant variance. The use of alternative therapies is determined by the type of patients, by thehuman capital of the hospital and by external/environment factors, hence cost is not alwaysan indicator of efficiency.3 To reduce the variance, the literature has proposed definingproducts using instruments such as DRGs, but these devices do not allow for definition ofa perfectly homogeneous product, given the variance arising from patients’ characteristics.In this context, pricing at the average cost might produce perverse effects on welfare, likeshutting down hospitals that are not necessarily inefficient. In the short run and from apurely welfare point of view, it is also necessary to ask whether penalties for being ineffi-cient such as closing the hospital are viable. Such a policy in fact affects the patients muchmore than the management of the hospital. The use of yardstick competition in health careshould follow the softer rules proposed by the public choice literature,4 and they might notbe optimal, as shown by Feld, Josselin and Rocaboy (2002).

As for quality, this is a variable whose definition is not unique since for some authorsit should be related to the appropriateness of the service while others stress its cost driverproperties. Most of the literature models quality as a cost driver and studies it in a con-text where hospitals are either sensitive to their own reputation or in a spatial competi-tion framework. In the former case, the quality of the care to be supplied, although notverifiable, becomes a relevant variable for the hospital in terms of its utility or activityvolume.5

In the latter case, suppliers compete for patients in a space (either physical or related tothe characteristics of their products); the patient can observe the quality of health care andchooses his supplier accordingly. In equilibrium, the optimal quality level depends on thenumber of suppliers and the reimbursement scheme offered; if the planner has sufficientdegrees of freedom, an optimal solution can always be reached (Gravelle, 1999). This

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literature mainly deals with the choice of patients as regards primary care, but it could beapplied to the choice of hospital care as suggested by Chalckley and Malcomson (1998).6

In the following sections a model will be developed in a framework where a collectivepurchaser (P) establishes a contract with two hospitals for the supply of hospital care ata specific price and patients choose where to receive health care in a spatial model a laHotelling using quality as a strategic variable.

3. The Model

The environment in which the collective purchaser P has to choose the contract can bedescribed as follows: N patients, uniformly distributed on a unit line, need hospital carethat is delivered through two hospitals, A and B, that are located at the extremes (0 and 1)of the line. Patients choose where to receive their treatment and can observe quality directlyor through an agent (their doctor) that acts in their own interest. The service is free at thepoint of use while traveling costs have to be borne by the patient.

The cost incurred by the hospital to produce health care is assumed to be a linear functionof quality, patients’ characteristics and the effort of the medical staff. The unit cost functioncan be written as:

Ci = βi + q − ei i = l, h (2)

where βi reflects severity of illness, ei is the effort of the medical staff and q is the qualitylevel.

In this paper we assume that the treatment offered is appropriate, hence quality can bedefined as a multidimensional vector that includes medical and non-medical variables whichaffect the outcome of health care and have a positive relationship with costs. In our modelwe assume that this relationship is linear.

βi is a random variable that reflects patients’ severity; to simplify the analysis it will beassumed that β can take only two values, βl for a patient with low severity and βh for apatient with high severity.7 Both events have a known probability equal to p and (1 − p)respectively.

The cost for taking care of a patient of type i is equal to β i + q, but it can be loweredthrough the effort ei of the management. The effort produces a disutility linear in the patientstreated, but increasing in the effort with positive second derivative i.e.

n f (e) > 0 n fe(e) > 0; n fee(e) > 0;

where subscripts denote derivatives. The hospital management participates in the productionprocess only if the reward received for each treatment, net of the production cost, producesa positive utility either in expected terms or in any state of the world:

U (ti − Ci − f (ei )) ≥ 0 PC1,

EU(ti − Ci − f (ei )) ≥ 0 PC2 (3)

where ti is the reimbursement scheduled and PC stands for Participation Constraint.The choice between the two participation constraints depends on the information structure

of the game and determines the risk-sharing properties of the contract. Although at the timethe contract is stipulated both parties have the same information on β, the hospital observesit before making its effort; in this case it will never accept a contract offering its reservationutility only in expected terms8 and the relevant constraint will be PC1. If β is observed afterthe decision on the effort, the contract in expected terms is a viable solution. In this case,the risk the hospital wishes to bear depends on the degree of risk-aversion, which in ourmodel is represented by the properties of U .

4. The Game

A two-stage game is played by the hospitals and the purchaser. In the first stage, a rangeof payment schemes to attain cost minimization and the optimal quality level is defined. Inthe second stage the purchaser anticipates hospital behavior and infers the amount it mustpay to induce hospitals to support the optimal quality level.

The model can be solved using backward induction. For effort determination, the secondstage of the game can be simply defined by the reservation utility so that the paymentscheme can be derived directly.

4.1. Payment Schemes and Effort Determination

Given the assumption on the cost and the utility function, it is possible to define the optimalcontract just for one patient and this solution can then be replicated for all the cases treated.At this stage of the game quality and cost are unknown to the purchaser, but the twoparameters are independent, i.e. the decision on whether to cheat does not depend on thelevel of quality that should be supplied. It is then possible to define the payment system fora set level q . A benchmark model where information is symmetric is now presented. Theproblem can be written as:

Min ptl + (1 − p)th

s.t. Ci = q + βi − ei i = l, hti − Ci − f (ei ) ≥ 0 PC1orEU (ti − Ci − f (ei )) ≥ 0 PC2

The choice between constraint PC1 and PC2 depends on the information structure of thegame9 and on the risk sharing attitudes of the two actors.

If nature reveals β before the hospital management has made its effort, PC1 is the relevantconstraint and the solution of the problem, presented in Appendix 1a, can be written as

fe(ei ) = 1 e∗l = e∗

= C∗i

+ f (e∗i)

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Table 1. Optimal contracts: Symmetric case.

PC1 PC2 (more likely solution)a

Effort conditions fe(ei ) = 1 fe(ei ) = 1

Utility in the optimal contract U (t∗l − C∗l − f (e∗

l )) = 0 p((t∗l − C∗l − f (e∗

U (t∗h

− C∗h − f (e∗

h )) = 0 + (1 − p)(t∗h

− C∗h − f (e∗

h )) = 0

Payment scheme ti = q∗ + βi − e∗ + f (e∗) t∗PPS = q∗ + E(β) − e∗ + f (e∗)

aWhen both parties are risk-neutral any solution that satisfies the budget constraint is possible. However,we might think that certainty in expenditure is preferred by P if this does not involve extra costs. Forthis reason P , being the Stackelberg leader of the game, might choose this solution.

It can be interpreted as follows: in any state of the world, the management is asked tomake an efficient, cost-minimizing effort in exchange for its reservation utility. This resultis the typical solution of a Stackelberg problem.

If the relevant constraint is PC2, the choice of the contract depends on the propensityto risk of the two agents. In the model it is assumed that P is risk neutral, i.e. it wants tominimize payment in expected terms. This assumption might be reasonable because of thenumber of cases to be treated and because the purchaser usually faces budget to control itstotal expenditure. The hospital, it might be risk neutral or risk averse. The optimal choicesfor P are presented in Appendix 1b and summarized in Table 1.

The first scheme corresponds to cost reimbursement, the typical way in which hospitalshave been financed in the past. Ellis and McGuire (1986) and other authors have widelycriticized this method of payment for its cost inflation properties and for not being ableto make hospitals produce an efficient level of care for their patients. They propose usinga prospective payment system instead. Prospective payment systems have a number ofdrawbacks, however: they can be used for large contracts, if the management is risk neutral,and—more important for our discussion—only when the hospital observes the type ofpatient after making its effort.

4.1.1. Asymmetry of Information. In this section we consider the effects of the intro-duction of asymmetry of information on the game described so far. The cost of hospitalcare, as much as the outcome in terms of improved health, depends on the severity of eachpatient which is observed only by the hospital and which is used by this agent to pursue themaximization of his objective function.10 The timing of information in the contract can besummarized as follows:

−−−−−−−→ −−−−−−−−−−−−−→ −−−−−−−−−→ −−−−−−−−−−−−→P sets contract only hospital observes β effort by hospital purchaser observes cost

The technology of production can be observed by both players; when the contract is signedboth parties share the same beliefs about the realization of β , but at some stage before theeffort is made the management can privately observe it. The cost of providing health careis observed by both parties ex post, unless the contract is a prospective payment scheme;effort and severity can be inferred from it. In this environment an asymmetry of information

arises and the hospital commands a rent on the private information (the exact realization ofβ) it can withhold. To stress the problems arising from asymmetry of information it will beassumed that the hospital is risk neutral, i.e. U (·) = K = 1.

If P offers a cost reimbursement scheme, the hospital will always produce health care ata cost Ch and ask for a payment equal to th If the patient is of type l, the management makesan effort elh = e∗

l − (βh − βl) and gets an extra utility equal to f (e∗h) − f (e∗

lh) but costs areinflated by �CC H = (e∗

l − e∗lh

). P has to change its strategy in the optimal reimbursementscheme if it wants to avoid the hospital playing an opportunistic game.

In this article we consider three pricing alternatives, namely prospective payment, incen-tive compatible schemes and yardstick competition. These solutions have different costs interms of incentives and also in terms of organization and control of the system. Below wewill concentrate on the costs in terms of rent that the hospital can command while discussionof the cost of control and design will be left to the following section.

4.1.1.1. Prospective Payment Schemes. We first analyze prospective payment systems asa method of solving the problem of information asymmetry.11 In this case, the hospital getsthe surplus (if any) and should be incentivated to perform all the necessary cost minimizingprocedures. This system can be effectively used only if the hospital accepts a contractwhere it receives its reservation utility expected terms, and this depends on the informationstructure of the game and the shape of the reservation utility function.

From an economic point of view, the first point is quite relevant: if the structure ofinformation allows the hospital to observeβ before making the effort, the use of a prospectivepayment inflates costs. The management in fact will ask for a prospective payment allowingit to get its reservation utility in the worst state; the problem for the purchaser can be writtenas:

Min ts.t. Ci = q + βi − ei i = l, h

ti − Ci − f (ei ) ≥ 0

The solution is characterized by the following conditions whose derivation is presentedin Appendix 2:

fe(eh) = 1 = e∗h

ePPSl = e∗

tPPS = Ch + f (eh) = t∗PPS + βh − E(β)

Ul = βh − βl > �U CH

Uh = 0

The cost paid by the purchaser in this case is in fact th , the same as in a cost reimbursementwith cheating. The management, on the other hand, is better off; it receives the samepayment, but since the purchaser cannot observe the cost, it can set the effort in the mostfavorable case at its optimal level.

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In this way it gets an extra utility for each low cost patient treated that is equal to thedifference between the cost for a high severity patient (βh) and its expected value (E(β)).

The hypothesis of the hospital being able to observe β (or to acquire better information)before treatment seems to be quite realistic and casts some doubts on the use of prospectivepayments in health care. In any case, even if we assume that the information structureallows for a reservation utility in expected terms, prospective payments can be used onlyfor large contracts or for homogeneous patients: in both cases in fact the average actual costwill be close to its expected value. When these conditions are not met, the risk of using aprospective payment system is that the hospital might go bankrupt even if it is efficient12

or it might promote the use of policies such as cream skimming and dumping.13

It is however important to note that the sustainability of this policy by the provider, whichin this model is not a Stackelberg leader, depends on its ability not to reveal to the purchaserany information on its actual cost. If this is not the case, as Ellis and McGuire (1986, p. 137)show, in the long run, the rent of the provider will disappear.

From a policy point of view another important consideration has to be made: in theprospective payment the ability of the patient to recover is not observed, even ex-post. Thisinformation is extremely valuable to P for designing incentive schemes in the future andalso for making its health policies more effective.

4.1.1.2. The Traditional Incentive Compatible Solution. When asymmetry of informationis explicitly considered, the problem for P can be written as:

s.t. Ci = q + βi − ei i = l, hti − Ci − f (ei ) ≥ 0ti − Ci − f (ei ) ≥ t j − C j − f (ei j )

where f (ei j ) represents the effort compatible with declaring C j when the true state of theworld is i. Two constraints characterize the problem: the first means, as before, that thehospital receives a reward producing at least its reservation utility; the constraint involvedis PC1 since it is assumed that the hospital can observe β before making its effort. Thesecond constraint, which is also called the incentive compatibility constraint, means thatthe hospital has the incentive to reveal truthfully the state of the world that has occurredand to make its effort accordingly. The solution is characterized by the following conditionswhose derivation is presented in Appendix 3:

fe(el) = 1 el = e∗l

fe(eh) = 1 − p[1 − fe(q + βl − Ch)] < 1 eh = eICh < e∗

= C∗l + f (el) + [

) − f(eIC

)] = Cminl + �U IC (7)

= CICh + f

) = Cminh + �CIC

Ul = f(eIC

) − f(eIC

) = �U IC

Uh = 0

The efficient level of effort is now required from the management only in the best stateof the world. If the patient is high morbidity, the hospital will deliver an effort shorter thanthe optimal one. This scheme allows compliance with the incentive compatibility constraintand the management receives its reservation utility only in the worst state of the world. Ifthe better state occurs, the hospital receives an extra payment that represents the rent for theinformation it commands and which is equal to �EU IC = p[ f (eIC

h ) − f (eIClh )]. The extra

cost deriving from an inefficient use of the effort in the worst state of the world inflatescosts by �CIC = (1 − p)[(e∗

h − eICh ) − (( f (e∗

h) − f (eICh ))].

The formula just presented can solve the problem of information asymmetry, but it opensup another important issue, i.e. setting the parameters of the incentive compatible contract,some of which might be difficult to translate into a sum of money.

This formula is quite similar to the system adopted in Italy to reimburse hospital care.After the reform, hospitals are reimbursed through a DRG-related system that shouldbe a prospective payment scheme. Italy has, however, introduced a correction through asupplementary payment to the hospital if the cost of the case is higher than a specificthreshold.

4.1.1.3. Information on β to Reduce Asymmetry of Information. Let’s assume that P canobserve a signal γ about the realization of β. The signal, which can assume two values, γl

and γh , is not perfect; its correlation with the true value of β is equal to ρ.In the classical model with asymmetry of information, the principal can use information

on the realization of the parameter it cannot observe (Rees, 1985) through a Bayesian updateof the probability of β after γ has been observed. The problem for P can then be writtenas:

Min zγ=i tl + (1 − zγ=i )th i = l, hs.t. Ci = q + βi − ei i = l, h

ti − Ci − f (ei ) ≥ 0ti − Ci − f (ei ) ≥ t j − C j − f (ei j )

where zγ=i is the conditional probability of a high/low recovery parameter given that theobserved signal was high/low. The solution for the game is similar to the one presented inAppendix 3 for the classical incentive compatible problem.

fe(el) = 1

fe(eh) = 1 − zγ=i [1 − fe(q + βl − Ch)] < 1

t ICSl

= C∗l + f (el) + [

f(eICS

) − f(eICS

)] = Cminl + �U ICS (9)

t ICSh

= C ICSh + f

) = Cminh + �C ICS

Ul = f(eICS

) − f(eICS

) = �U ICS

Uh = 0

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It is interesting to note the role played by z in determining the two incentive schemes. zin fact assumes the following two values:

z A=l = p + (1 − p)ρ

z A=h = p(1 − ρ)

In the first case, having observed γl , βl is more probable than βh , eh is reduced, C ICSh

increases, the information rent is reduced and the inefficiency cost increases since the eventl is more probable.

In the second case, βh increases its probability of occurring and for this reason, the effortin this state is increased, hence increasing the efficiency of the game. Of course, the cost isreduced and the incentive for the first occurrence is increased, but this event should occurwith a lower probability. The use of information on β reduces the information rent of thehospital, but does not avoid the problem of cheating.

4.1.1.4. Yardstick Competition/Benchmarking. Yardstick competition is an instrument thatthe literature studying pricing policies for public utilities uses widely to reduce the rentarising from asymmetry of information. The environment in which these models are set isone where the regulator has to define a pricing policy for several public utilities that do notcompete in the product market because each of them has a spatial monopoly and it usuallyimplies fixing the price to the average cost of the industry. However, this system can besuccessfully applied only if we are prepared to accept that firms can go bankrupt or thatthey can have a deficit in the short run. These two conditions might not easily apply to thehealth care market.

The structure of yardstick competition that is most suitable for hospital care is a sortof benchmarking based on the observed cost for the competitors in the same market. Inthis context, asymmetry of information plays a fundamental role because costs can beeasily audited, but patients’ severity remains private information for the hospital since thisinformation is difficult or too expensive to be tracked down by the purchaser. This approachis studied by the public choice literature14 and can be described as follows:

Period 1: A and B declare their costs and the level of quality qi and the purchaser derivesβ. For ease of exposition, it will be assumed that β represents local characteristics ofthe population to be treated and that individuals are homogeneous with respect to thisparameter whose realization has then to be derived only once.

Period 2 to N: P decides whether to renew the contract15 on a relative performance evaluationof the two hospital. If the management is confirmed, stage one is repeated.

Period N + 1: The management is not reconfirmed irrespective of what is done in N .

To develop our model we will use the same basic assumptions presented in Section 2as regards the cost function, the utility of the hospital and patients’ severity. It will also beassumed that β is not correlated among periods and that all the other costs and functionsare fixed through time.

βs are specific to each hospital, but there is a degree of correlation r among them, so thatthe observation of a parameter in hospital A makes the realization of the same level moreprobable in B. In particular, we can define the joint probabilities of the event βlβh as:

π(β i

h, βj

) = (1 − p)(1 − p(1 − r ))

π(β i

l , βj

) = p[(1 − (1 − p)(1 − r )] (10)

π(β i

l , βj

) = π(β i

h, βj

) = p(1 − p)(1 − r )

An opportunistic behavior can be defined as follows: if the patient is low cost (βl is thetrue state of the world) the hospital decreases its effort (so that the observed cost is equalto Ch) and gets an extra rent equal to �EU C H . This behavior can be repeated throughtime if the management is confirmed, something that depends on the value of β in A andB and on the behavior of both hospitals. In period n, given that the management will notbe reconfirmed, P knows that the agent will cheat with certainty. To avoid this cost, anincentive compatible scheme is used so that the expected payoff in the last period is equalto �EU IC.

There is no co-ordination among the hospitals, i.e. the decision of each of them onwhether to cheat is the solution to a straightforward Nash game. To simplify the expositionwe present the results of a two-period game. Each hospital has just two strategies (cheat ornot cheat); there are four possible outcomes for the game that are derived in appendix fourand are presented in Table 2.

Table 2 shows that the opportunistic outcome is always a Nash equilibrium unless p = 0.This result depends on the expected payoff received by the hospital. The strategy of revealingthe truth in the first period is not always rewarded with re-confirmation, and cheating, evenwhen it is a unilateral decision, might not be punished. This produces a negative incentiveto truth revelation. To make the hospital play in a non-opportunistic way, it is necessary toincentivate the management to produce the optimal state-contingent effort either by usingfines if the cheating is discovered or by rewards if the truth is told. In the first case we canobserve that if a fine equal to [�EU C H −�EU ICσ p(1−(1− p)(1−r ))] were imposed, notcheating might become a weak Nash equilibrium.16 The results of this section do not dependon the discount factor σ , on the time horizon chosen or on the application of restrictions tocheating in the last period.

Table 2. The possible outcomes of the game.

Not cheat Cheat

A Not cheat A: σ [1 − p(1 − p)(1 − r )]�EU IC A: σ�EU IC

B: σ [1 − p(1 − p)(1 − r )]�EU IC B: [�EU C H + σ (1 − p)�EU IC]

Cheat A: [�EU C H + σ (1 − p)�EU IC] A: (�EU C H + σ�EU IC)

B: σ�EU IC B: (�EU C H + σ�EU IC)

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4.2. The Determination of Quality by the Hospital: The Hotelling Game

For hospital care, quality is assumed to be non-verifiable, hence its level cannot be directlycontracted. In this paper, we assume that quality can be indirectly controlled through thechoice of patients.

In the health care market, patients do not usually pay for treatments at the point of useand the choice of the hospital depends on perceived quality and the cost incurred in movingto the provider’s location. For this reason, in this model we assume that in the second stageof the game the two hospitals compete for patients through quality using the rules of spatialcompetition.

The population requiring health care consists of N patients, uniformly distributed on aunit line while the two hospitals are located at the two extremes (0 and 1).

Patients have the same valuation of quality characteristics and incur the same marginaldistance cost s. The utility function of a patient located at point x on the line of unit lengthcan be written as:

Vx ={ϕqa − sx if patient is admitted to hospital A

ϕqB − s(1 − x) if patient is admitted to hospital B

where ϕqA is the monetary equivalent gain derived from the use of hospital services ofquality qA from hospital A and sx and s(1− x) are travel costs. The patient opts for hospitalcare if he receives at least a non-negative utility, i.e. Vx ≥ 0.

Patients are indifferent between hospital A and hospital B when:

ϕqA − sx = ϕqB − s(1 − x)

This expression can be written in terms of the location of patients as follows:

x = ϕ(qA − qB)

and the demand for hospital i can be obtained multiplying the distance by the density which,given the unit length of the line, is equal to N . The demand for each hospital can then bewritten as:

D j =[ϕ(qi − q j )

]N (12)

Each hospital is a competitor with the other hospital for the demand within the locationof the two outlets and wants to maximize its total utility:

Max[p(tl − C∗l − f (e∗

l )) + (1 − p)(th − C∗h − f (e∗

h))] ∗ D j (13)

Given that the rules for cost reimbursement have already been defined, we can rewritethe following expression as:

Max (q∗ − q j + �K )) ∗ D j (14)

where �k is the information rent the hospital enjoys. Its value depends on the environmentin which the hospital competes.

For the ICC solution, the evaluation of the information rent is quite straightforward. Foryardstick competition and prospective payment schemes the problem is more complicated.The information rent could in fact be used to increase demand, but it would be a signaleither of cheating (yardstick competition) or of cost of production (prospective paymentscheme). In both cases, it thus seems more reasonable to assume that the hospital does notuse the information rent to compete. In our model, �K assumes the following values:

�K =

0 for the symmetric case

0 for yardstick competition and prospective payment

�EU IC for the ICC case

�EU ICS for IC with signal

The F.O.C. can be written as:

−D j + (q∗ − q j + �K ) ∗ ϕN

Solving for q we can write:

q j = 1

(q∗ + qi + p�K − s

With identical hospitals, it seems reasonable to assume that a symmetric Nash equilibriumexists in which firms choose the same quality.17 The quality for each competitor can thenbe written as:

q j = q∗ + �K − s

ϕ(16)

The hospital reduces the quality paid by the purchaser by the amount s/ϕ, which canbe interpreted in terms of a spatial monopoly rent. Since going to the hospital is costly,each of them enjoys a position rent that does not allow the market to be fully competitive.The second important element in (16) is the trade-off between quality and opportunisticbehavior. The information rent is in fact used to compete for quality with the other hospitaland attract patients.18

340 LEVAGGI

4.3. The Choice of the Optimal Quality Level by the Purchaser

The purchaser chooses q∗ in order to minimize costs while inducing all the patients needingtreatment to be admitted. This condition requires the marginal patient to get zero utilityfrom health care. Hospitals share the market equally, hence the marginal patient is locatedat x = 1/2.

The quality to be reimbursed to make the marginal patient go to hospital can be determinedas follows:

ϕq j − s

2= 0 q j = s

2ϕhence : q∗ = 3s

2ϕ− �K

The quality incentive that P has to supply is directly related to the hospital’s rent from itscheating. This is an important result: the information rent is fully recovered in the secondstage of the game through quality setting. This does not mean that all the contracts have thesame cost since asymmetry of information has two dimensions that are important here: theinformation rent and the inefficient use of resources.

5. Comparing the Different Solutions

In this section the expected cost for the different contracts will be presented using thesolution for the game with symmetric information as a benchmark. Table 3 presents thetotal cost (T ) that P incurs to get the N patients treated.

Let’s start by examining the first three models. The Hotelling game played at the secondstage allows P to get back, in the form of higher quality, the information rent paid to thehospital. The benchmark contract is still cost minimizing since in this case the effort isused in the best possible way. In the other forms, the effort falls below its optimal level andthe production cost is higher than it should be. The comparison between ICC and the othersecond best contracts is not straightforward given that we have to compare inefficiency witha rent. By using the first order conditions for the effort, it is however possible to determinethat T IC < T C H . Yardstick and prospective payment contracts are the most inefficient; forICC, in fact, the effort falls below its optimal level in the worst state of nature while foryardstick the effort is reduced in the best state of nature in order to report the highest cost.For prospective payment system, the effort made by the hospital is cost minimizing, butnone of the saving is shared with the purchaser.

Table 3. Total cost of providing health care.

Benchmark Yardstick/PPS ICC ICS

t∗l = C∗l + f (e∗) tl = th = tPPS=C∗

h + f (e∗h ) t IC

l = C∗l + f (e∗

l ) + �IC t ICSl = C∗

l + f (e∗l ) + �ICS

t∗h = C∗h + f (e∗) t IC

h = CICh + f

)t ICSh = C DM

h + f(eICS

q∗ = 3s2ϕ

− �IC q∗ = 3s2ϕ

− �ICS

T MIN = N[ 3s

2ϕ+ ECmin T C H = N

[ 3s2ϕ

+ Ch + f (e)]

T IC = N[ 3s

2ϕ+ E(CIC) T ICS = N

[ 3s2ϕ

+ E(C ICS)+ f (e)

] +E( f (eIC))] +E( f (eICS))

The contract with information permits a further reduction in the expected cost as onewould expect. From a policy point of view, it is interesting to ask where P might obtainsignals about the realization of β. One possible way would be to manage one of the hos-pitals and get to know the true realization of this parameter. This information could thenbe used to infer the β for the independent hospital. This framework has interesting pol-icy implications as regards the objective function of the hospitals, and the managementcost efficiency incentives. It creates a form of mixed oligopoly that needs to be assessedseparately (Levaggi, 2004). Another way to get information on β is through its past re-alization and epidemiological data on the population. The observation of β over time isan important indicator for the purchaser of the needs of his population and might permitreduction of the hospital information rent. Knowledge of the true parameter is anywayvaluable for forecasting future needs and future costs. This is another reason why prospec-tive payment contracts are not an optimal instrument in health care. They do not allow thepurchaser to know even ex post the true health status of his population and in this frame-work, establishing health care priorities for the future is more complicated than it shouldbe.

6. Conclusions

The peculiar characteristics of health care make it difficult for the market forces to workefficiently. The article sets up a model combining the rules of spatial competition a laHotelling with several contracts to define a scheme for finding an efficient (cost minimiza-tion) mechanism to reimburse hospital care. Consumers can observe the quality of the carethey receive and choose to go to the hospital that maximizes the difference between qualityand traveling cost. Hospitals’ behavior is determined by maximization of the utility of theirmanagement, which in turn depends on the revenue they receive minus costs. Costs dependon the quality of the service delivered, on the effort of the managerial staff and on the abilityof the patient to recover quickly. This information is private to the hospital and determinesasymmetry of information.

The model presented in this paper shows that even if we assume that quality can beindirectly controlled through the behavior of patients, designing a contractual form thatpermits minimization of costs is rather complicated. The problem arises from the presenceof asymmetry of information as regards the cost for health care and its quality. Health careis in fact an input itself in the process of a patient’s recovery and the relationship betweenhealth care and health gains is not observable.

From the model it emerges that prospective payments and yardstick competition, at leastin the way they can be used in health care, have poor performances, and a straight incentivecompatible contract seems to be preferred in this case.

Asymmetry of information increases costs due to an inefficient use of resources andto the information rent the hospital receives. If hospitals compete for patients in a spatialmodel, the information rent can be recovered in the form of higher quality, but the inefficientuse of resources cannot be avoided. An incentive compatible scheme permits reduction ininefficiency and even if it involves a higher payment for quality, the total cost is lower thanunder other schemes.

342 LEVAGGI

The information on the actual morbidity level of the population that the hospital revealsonly with an ICC scheme might have a value per se: this parameter might have a correlationwith future morbidity levels (hence costs) and it might also be used as an index to evaluatethe impact of the purchaser’s activity on the level of health of the reference population.

This conclusion has important policy implications. The implementation of PPS and yard-stick competition models is quite straightforward while the use of incentive-compatibleschemes implies the knowledge or the acquisition of information on the money evaluationof utility-related parameters. The focus of the research in this sector is then shifted towardsthe actual implementation of these formulas.

The results of the paper can be extended in several directions; non-linear cost and utilityfunctions could be adopted as well as forms of imperfect observation of the quality bypatients. Both elements might weaken the relationship between information rent and qualitybut would not alter the basic results of the analysis which has proved to be robust tointroducing quality in the cost function as a multiplicative element or in a square form.Another interesting line of research would be to investigate the social welfare properties ofthe different contracts that have different productivity levels in terms of quality.

Appendix 1a

The problem faced by P can be written as:

s.t. Ci = q + βi − ei i = l, hti − Ci − f (ei ) ≥ 0

P acts as a Stackelberg leader, the inequality for the reservation utility of the hospitalcan be taken as an equality. The two constraints can then be substituted in the objectivefunction giving:

Min Tr = p(q + βl − el + f (el)) + (1 − p)(q + βh − eh + f (eh))

The F.O.C. for the problem can be written as:

∂T R

∂elp[ fe(el) − 1] = 0

∂T R

∂eh(1 − p)[ fe(eh) − 1] = 0

Giving:

fe(el) = fe(eh) = 1

t∗l = f (e∗

l ) + C∗l

t∗h = f (e∗

h) + C∗h

The S.O.C can be written as:

∂2T R

∂e2l

p fee(el) > 0

∂T R

∂e2h

(1 − p) fee(eh) > 0

Appendix 1b

The problem faced by P can be written as:

s.t. Ci = q + βi − ei i = l, hpU (tl − Cl − f (el)) + (1 − p)U (th − Ch − f (eh)) ≥ 0

As before, the inequality for the reservation utility of the hospital can be taken as anequality. The first constraints can then be substituted in the second constraint. The problemcan be solved using the Lagrange approach:

Min T r = p(tl) + (1 − p)(th) − λ[pU (tl − Cl − f (el))

+ (1 − p)U (th − Ch − f (eh))]

∂T R

∂elλp

[fel (el) − 1

∂T R

∂ehλ (1 − p)

[feh (eh) − 1

∂T R

∂tlp − λpUtl

∂T R

∂th(1 − p) − λ(1 − p)Uth

giving:

fel (el) = feh (eh) = 1

Utl = Uth

pU (tl − Cl − f (el)) + (1 − p)U (th − Ch − f (eh)) = 0

where Ut is the marginal utility of the reimbursement scheme in any state of the world.

344 LEVAGGI

The Bordered Hessian can be written as:

λp fel el (el) 0 0 0∂2T R

∂e2l

0 λ (1 − p) feh eh (eh) 0 0∂2T R

∂e2h

λpUtl fel (el) 0 −λpUtltl 0∂2T R

∂t2l

0 λ(1 − p)Uth feh (eh) 0 −λ(1 − p)Uthth

∂2T R

∂t2h

H1 = λp fel el (el) > 0

H2 = λ(1 − p) feh eh (eh) ∗ H1 > 0

H3 = −λpUtltl ∗ H2 > 0

H4 = −λ(1 − p)Uthth ∗ H3 > 0

Appendix 2

Min ts.t. Ci = q + βi − ei i = l, h (i)

t−Cl − f (el) ≥ 0 (ii)t−Ch − f (eh) ≥ 0 (iii)

In this context (iii) is binding and the problem can be solved for eh

feh (eh) = 1

tPPS = Ch + f (eh)

Uh = 0

The hospital will then set el to maximize:

Max tPPS − Cl − f (el)

Giving:

fel (el) = 1

Ul = tPPS − Cl + f (e)

Ul = βh − βl

Appendix 3

s.t. Ci = q + βi − ei i = l, hti − Ci − f (ei ) ≥ 0ti − Ci − f (ei ) ≥ t j − C j − f (ei j )

The problem has to be solved in terms of observable variables such as the cost and not interms of effort e that only the hospital can observe. From the first constraint we can deriveei = q + βi − Ci . The third constraint in the problem, the so-called Incentive CompatibleConstraint, can be written as:

tl − Cl − f (q + βl − Cl) ≥ th − Ch − f (q + βl − Ch)

th − Ck − f (q + βh − Ch) ≥ tl − Cl − f (q + βh − Cl)

The SOC on the disutility of the effort allow to conclude that the second inequality isalways satisfied. Let’s now observe the first inequality. It states that the net payment to thehospital in the best states of the world has to be at least equal to the payment received inthe worst state of the world plus a compensation for the disutility of the effort. Let’s nowobserve the participation constraint. In the worst state of the world, the hospital receives acompensation that is equal to th = Ch + f (q +βh − Ch). Let’s now observe the l.h.s. of theequation. We can observe that f (q+βl −Ch) < f (q+βh −Ch), hence the utility received inthe best state of the world is greater than zero, which in turn means that the first participationconstraint is always satisfied. With all this in mind we can write the minimization problemas:

Min TR = p{Cl + f (q + βl − Cl) + [ f (q + βh − Ch) − f (q + βl − Ch)]}+ (1 − p)[Ch + f (q + βh − Ch)]

∂Cl= p[1 − fCl (q + βl − Cl)]

∂Ch= p[ fCh (q + βl − Ch) − f ′(q∗ + βh − Ch)] + (1 − p)[1 − fCh (q + βh − Ch)]

giving as solution:

fCl (q + βl − Cl) = 1

fCh (q + βh − Ch) = 1 − p[1 − fCh (q + βl − Ch)] < 1

t ICl = C∗

l + f (q + βl − C∗l ) + [

f(q + βh − CIC

) − f(q + βl − CIC

346 LEVAGGI

t ICh = C∗∗

h + f(q + βh − CIC

Ul = f(q + βh − CIC

) − f(q + βl − CIC

Uh = 0

The IC reimbursement scheme can be written as:

t ICl = C∗

l + [f(q + βh − CIC

)] + f (q + βl − C∗l ) − f

(q + βl − CIC

t ICh = C∗

h + f(q + βh − CIC

Appendix 4

(a) Neither hospitals cheat in period oneBoth hospitals apply the cost minimizing effort in the first period and cheat in the second.However, due to imperfect correlation among βs, the hospital management might notbe reconfirmed with the following probability:

φi = 1 − π(β i

h, βj

) = 1 − p(1 − p)(1 − r ) ≤ 1

Being honest does not secure renewal of the contract and this is a serious disincentiveto revealing the true state of the world. The payoff for both hospitals can be writtenas:

E(pay) = 0 + σ [1 − p(1 − p)(1 − r )]�EU IC

where σ is the discount factor for the rent in the second period.19

(b) Both hospitals cheat in period oneCheating in period one consists of making the optimal, cost minimizing effort only if thepatient is high morbidity. If the patient is low severity, the effort is decreased accordinglyso that the observed cost will be equal to Ch . Since both hospitals announce the sameβ, both managements will be confirmed; the expected payoff for both hospitals can bewritten as:

E(pay) = (�EU C H + σ�EU IC)

(c) Just one of the two hospitals cheatsLet’s assume that hospital A cheats and produces treatments at a cost equal to Ch .The hospital receives a higher utility in the first period with probability p, but themanagement might not be re-confirmed if the other hospital produces care at cost Cl .

The probability of being reconfirmed is then equal to:

φi = π(β i

h, βj

) + π(β i

l , βj

) = 1 − p

and the expected payoff is equal to:

E(pay) = �EU C H + σ (1 − p)�EU IC

Hospital B reveals the true cost of its patients in the first period and, since hospi-tal A always declares that its patients are high cost, the management will always bereconfirmed. The payoff for hospital B can then be written as:

E(pay) = 0 + �EU IC

Notation

N Number of patients needing hospital careA Hospital AB Hospital Bs Transport costx DistanceCi Cost of providing health care to patient of type iβi Factor affecting the ability of patients to take advantage of health careei State contingent effort of the managementq Quality of health care offeredU Utility of the hospital managementti State contingent reimbursement schemep Probability that the patient is low morbidityr Correlation coefficient between β A

i and βBi

π Probability of events correlated with shocksφ Probability of the management being reconfirmedγi Signal on β

ρ Correlation coefficient between γi and βi

z Update of the probability of βi given that observed γ is γ j

Acknowledgments

The author would like to thank F. Menoncin, E. Minelli and the referees for their helpfulcomments which have greatly improved the paper. The usual disclaimer applies. Financialsupport form PRIN n. 2003130120 004 is gratefully acknowledged.

348 LEVAGGI

1. See Enthoven (2002), Chalkley and Malcomson (2002), Dranove, Shanley and Simon (1992), Kessler andMcCleallan (2000), Gaynor and Vogt (2003), Biglaiser and Ma (2003)

2. Gravelle (1999), Levaggi and Rochaix (2004) show improvements while Levaggi and Moretto (2004) showa case where patients’choices might reduce welfare.

3. In the U.K. doctors in training can be used to get labour at a lower cost; the presence of a well-developedhome care service allows patients to be discharged earlier.

4. See Shleifer (1985), Salmon (1987), Besley and Case (1995), Feld, Josselin and Rocaboy (2002).5. See Chalkley and Malcomson (2000), Levaggi and Moretto (2004)6. Proposition 2 of their paper suggests that hospitals compete for patients at a quality level.7. It is assumed that morbidity is correlated with the recovery speed of the patient and hence with cost. If

morbidity is low, the recovery rate is high and hence cost is low.8. See Sappington (1983) and Levaggi (1996).9. Both actors observe β before or after the provider has made its effort.

10. See, for example, Levaggi (1999).11. For a review see Newhouse (1996) and Chalkley and Malcomson (2000).12. Or it might have a profit and be inefficient.13. The effect of these policies is presented in Ellis (1997), Lewis and Sappington (1999), Levaggi and Montefiori

(2005).14. See Feld, Josselin and Rocaboy (2002).15. If the provider is a government agency, it might have the power to change the management of the hospital.16. An alternative incentive is to let the management of the “efficient” hospital run both hospitals in the second

period. It can be shown that this incentive motivates only a hospital with a very low share of the market toreveal the truth in the first period. However, given that hospitals are equal, in this model patients will beequally distributed as shown in the following section of the paper.

17. See Gravelle(1999)18. The hospital uses all the information rent for increasing quality in this model due to the assumption of

patients’ linear utility as pointed out by one of the referee. In this case, given that in the other incentiveschemes considered the information rent is not used to increase quality, this assumption does not affect theordering of the costs and it allows to derive a precise cost function.

19. For hospital B the same procedure can be used and the same result is achieved.

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Hospital Health Care: Pricing and Quality Control in a Spatial Model with Asymmetry of Information

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